Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev-Petviashvili equation with p-power

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1 + 1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solution with arbitrary functions for the PKPp.     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

   Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

Amplitude Modulation of Nonlinear Waves in Fluid-Filled Tapered Tubes

We study the modulation of nonlinear waves in fluid-filled prestressed tapered tubes. For this, we obtain the nonlinear dynamical equations of motion of a prestressed tapered tube filled with an incompressible inviscid fluid. Assuming that the tapering angle is small and using the reductive perturbation method, we study the amplitude modulation of nonlinear waves and obtain the nonlinear Schrödinger equation with variable coefficients as the evolution equation. A traveling-wave type of solution of such a nonlinear equation with variable coefficients is obtained, and we observe that in contrast to the case of a constant tube radius, the speed of the wave is variable. Namely, the wave speed increases with distance for narrowing tubes and decreases for expanding tubes.     

Kadomtsev-Petviashvili方程的精确解

齐次平衡法是把非线性偏微分方程转换成带约束条件的线性偏微分方程的一种很好的方法 .本文在齐次平衡法的基础上具体讨论了KP方程的精确解 ,包括孤波解 ,一般的行波解 ,有理函数解和一种新类型的解 .     

A nonlinear nonstandard finite difference scheme for the linear time-dependent Schrödinger equation

We state and study the various limiting forms and their associated mathematical properties of a nonlinear finite difference scheme for the linear time-dependent Schrödinger partial differential equation (PDE). A formal solution to the full equation is given.     

一类非线性脉冲泛函微分方程解的渐近性

利用构造Lyapunov泛函方法讨论了一类具有正负系数的非线性脉冲泛函微分方程解的渐近性,给出方程的解趋于一个正常数的充分条件,并举例说明定理条件和结论的可实现性.     

Existence-uniqueness result for a nonlinear n-term fractional equation

We prove the existence-uniqueness of the solution to the nonlinear n-term time-fractional differential equation with constant coefficients in the Banach space C([0,T]),

(1)     

Probabilistic Solutions for a Class of Path-Dependent Hamilton-Jacobi-Bellman Equations

This article shows that the solution of a backward stochastic differential equation under G-expectation provides a probabilistic interpretation for the viscosity solution of a type of path-dependent Hamilton-Jacobi-Bellman equation. Particularly, a G-martingale can be considered as a nonlinear path-dependent partial differential equation (PDE). We also show that certain class of path-dependent PDEs can be transformed into classical multiple state-dependent PDEs. As an application, the path-dependent uncertain volatility model can be described directly by path-dependent Black-Scholes-Barrenblett equations.     

Probabilistic solutions to nonlinear fractional differential equations of generalized Caputo and Riemann–Liouville type

This paper provides well-posedness and integral representations of the solutions to nonlinear equations involving generalized Caputo and Riemann–Liouville type fractional derivatives. As particular cases, we study the linear equation with non constant coefficients and the generalized composite fractional relaxation equation. Our approach relies on the probabilistic representation of the solution to the generalized linear problem recently obtained by the authors. These results encompass some known cases in the context of classical fractional derivatives, as well as their far reaching extensions including various mixed derivatives.     

一类二阶变系数线性微分方程的积分因子解法

通过寻求积分因子f1(x),f2(x),求解一类二阶线性微分方程,包括二阶常系数线性微分方程和二阶Euler方程.     

Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator

We study an abstract nonlinear evolution equation governed by a time-dependent operator of subdifferential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.     

一阶中立型时滞差分方程振动的充分必要条件

研究了一阶常系数中立型时滞差分方程A[x(n)-px(n-τ)]+qx(n-σ)=0的振动性.通过构造若干适当的函数,分别得到了在01两种情况下该方程的一切解振动的充分必要条件.     

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev′e Equation: an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.     

一类二阶变系数线性微分方程的解法

介绍如何通过变换把二阶变系数线性微分方程转化为一阶非线性微分方程,进而利用待定系数法对其求解,并对二阶变系数线性微分方程与一阶常系数非线性微分方程的内在的关系进行讨论.     

Ibragimov-type invariants for a system of two linear parabolic equations

We obtain new semi-invariants for a system of two linear parabolic type partial differential equations (PDEs) in two independent variables under equivalence transformations of the dependent variables only. This is achieved for a class of systems of two linear parabolic type PDEs that correspond to a scalar complex linear (1 + 1) parabolic equation. The complex transformations of the dependent variables which map the complex scalar linear parabolic PDE to itself provide us with real transformations that map the corresponding system of linear parabolic type PDEs to itself with different coefficients in general. The semi-invariants deduced for this class of systems of two linear parabolic type equations correspond to the complex Ibragimov invariants of the complex scalar linear parabolic equation. We also look at particular cases of the system of parabolic type equations when they are uncoupled or coupled in a special manner. Moreover, we address the inverse problem of when systems of linear parabolic type equations arise from analytic continuation of a scalar linear parabolic PDE. Examples are given to illustrate the method implemented.     

A new approach for solving highly nonlinear partial differential equations by successive differentiation method

In this work successive differentiation method is applied to solve highly nonlinear partial differential equations (PDEs) such as Benjamin–Bona–Mahony equation, Burger's equation, Fornberg–Whitham equation, and Gardner equation. To show the efficacy of this new technique, figures have been incorporated to compare exact solution and results of this method. Wave variable is used to convert the highly nonlinear PDE into ordinary differential equation with order reduction. Then successive differentiation method is utilized to obtain the numerical solution of considered PDEs in this paper. Copyright © 2017 John Wiley & Sons, Ltd.     

Summability of Formal Power-Series Solutions of Partial Differential Equations with Constant Coefficients

We study Gevrey properties and summability of power series in two variables that are formal solutions of a Cauchy problem for general linear partial differential equations with constant coefficients. In doing so, we extend earlier results in two articles of Balser and Lutz, Miyake, and Schäfke for the complex heat equation, as well as in a paper of Balser and Miyake, who have investigated the same questions for a certain class of linear PDE with constant coefficients subject to some restrictive assumptions. Moreover, we also present an example of a PDE where the formal solution of the Cauchy problem is not k-summable for whatever value of k, but instead is multisummable with two levels under corresponding conditions upon the Cauchy data. That this can occur has not been observed up to now.     

A Power Penalty Approach to Numerical Solutions of Two-Asset American Options

This paper aims to develop a power penalty method for a linear parabolic variational inequality (Ⅵ) in two spatial dimensions governing the two-asset Ameri-can option valuation. This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constant λ>1 and a power parameter k>0. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order (λ<-k/2>). A fitted finite volume method is designed to solve the nonlinear PDE, and some numerical experiments are performed to illustrate the usefulness of this method.     

五阶势MKdV方程的一个不变性

§1Introduction Asweknow,Backlundtransformation[1-3]isaverypowerfulwayforfindingnonline evolutionequations.Inrecentdecades,Painlevéanalysis[4]hasbecomeaverypopu methodtoobtainBacklundtransformation.Inreference[5],fordevelopingthetheory Painlevéanalysis,AndrewPickeringintroduceanewexpansionvariableZwhichsatisf thefollowingRicattiSystem:Zx=1-AZ-BZ2,Zt=-C+(AC+Cx)Z-(D-BC)Z2.(1.Astheapplicationofthenewexpansionvaraible,thepotentialfifth-orderMKd equation(PMKdV5)-vxt+(vxxxxx-10k2v2…